Exponents and roots

The rules come from counting factors

$a^n$ means $n$ factors of $a$ multiplied together. Every exponent rule is a way of counting those factors.

Rule	Why
$a^m \cdot a^n = a^{m+n}$	$m$ factors and then $n$ more
$\frac{a^m}{a^n} = a^{m-n}$	$n$ factors cancel
$(a^m)^n = a^{mn}$	$n$ groups of $m$ factors
$(ab)^n = a^n b^n$	each factor of $ab$ splits in two
$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$	the same, for a quotient

The quotient rule explains the definitions that look odd at first. $\frac{a^3}{a^3}$ is 1, and the rule gives $a^0$, so $a^0 = 1$ for any non-zero $a$. $\frac{a^2}{a^5} = \frac{1}{a^3}$, and the rule gives $a^{-3}$, so a negative exponent means a reciprocal: $2^{-3} = \frac{1}{8}$. A negative exponent never makes the number negative.

There is no rule for sums. $2^3 + 2^4$ is $8 + 16 = 24$, which is not a power of 2. What you can do is factor: $2^3 + 2^4 = 2^3(1 + 2) = 3 \cdot 2^3$. In the same way, $5^{10} + 5^{10} + 5^{10} + 5^{10} + 5^{10} = 5 \cdot 5^{10} = 5^{11}$.

Work on a common base

Most exponent questions become easy once every power is written on the same base, usually a small prime. Learn the common ones: $4 = 2^2$, $8 = 2^3$, $16 = 2^4$, $32 = 2^5$, $9 = 3^2$, $27 = 3^3$, $81 = 3^4$, $25 = 5^2$, $125 = 5^3$.

With a common base you can simplify by adding and subtracting exponents, and you can solve equations: if $3^p = 3^q$ then $p = q$ (for a base other than 0, 1 and $-1$).

To compare two powers, make either the bases or the exponents match. $2^{30}$ and $3^{20}$ share a factor 10 in their exponents: $2^{30} = (2^3)^{10} = 8^{10}$ and $3^{20} = (3^2)^{10} = 9^{10}$, so $3^{20}$ is larger. No calculation of a ten-digit number is needed.

Roots

$\sqrt{a}$ is the non-negative number whose square is $a$. So $\sqrt{25} = 5$, not $\pm 5$, even though the equation $x^2 = 25$ has the two solutions $5$ and $-5$. In the same spirit, $\sqrt{x^2} = |x|$.

Roots are fractional exponents: $\sqrt{a} = a^{1/2}$, $\sqrt[3]{a} = a^{1/3}$, and $a^{m/n} = \left(\sqrt[n]{a}\right)^m$. So $8^{2/3} = 2^2 = 4$, and $16^{-1/2} = \frac{1}{4}$.

Roots respect multiplication and division, but not addition:

$$\sqrt{ab} = \sqrt{a}\sqrt{b}, \qquad \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}, \qquad \sqrt{a + b} \ne \sqrt{a} + \sqrt{b}$$

(for non-negative $a$ and $b$; the last inequality fails only when $a$ or $b$ is 0). A quick reminder: $\sqrt{9 + 16} = 5$, while $\sqrt{9} + \sqrt{16} = 7$.

To simplify a root, pull out the largest square factor: $\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}$. Roots with the same radicand combine like terms: $3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2}$, but $\sqrt{2} + \sqrt{3}$ does not simplify. To remove a root from a denominator, multiply top and bottom by it: $\frac{6}{\sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3}$.

Size: where the number sits matters

Raising a number to a power does not always make it bigger.

• For $x > 1$: $\sqrt{x} < x < x^2 < x^3$.
• For $0 < x < 1$: $x^3 < x^2 < x < \sqrt{x}$. Squaring $\frac{1}{2}$ gives $\frac{1}{4}$, smaller; its square root is about 0.71, larger.
• For $x = 0$ or $x = 1$, all of these are equal.
• For a negative base, even powers are positive and odd powers negative: $(-2)^4 = 16$, $(-2)^3 = -8$.

Watch the brackets: $-3^2$ means $-(3^2) = -9$, while $(-3)^2 = 9$.

These facts decide many comparison questions. When a stem says $0 < x < 1$, it is usually inviting you to assume powers grow; they shrink.

Benchmarks worth knowing

Knowing $\sqrt{2} \approx 1.41$, $\sqrt{3} \approx 1.73$ and $\sqrt{5} \approx 2.24$, and the squares up to $15^2 = 225$, lets you estimate fast. The on-screen calculator in the Quantitative Reasoning section can check arithmetic, but the rules above are usually quicker than keying in long powers.